Fractional-power approach for solving complete elliptic abstract differential equations with variable-operator coefficients

This work is devoted to the study of a complete abstract second-order differential equation of elliptic type with variable operators as coefficients. A similar equation was studied by Favini et al [6] using Green's kernels and Dunford functional calculus. Our approach is based on the semigroup theory, the fractional powers of linear operators, and the Dunford's functional calculus. We will prove the main result on the existence and uniqueness of a strict solutions using combining assumptions from Yagi [16], Da Prato-Grisvard [3], and Acquistapace-Terreni [1].